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KoMaL A Problems
KoMaL A Problems 2023/2024
A. 862
A. 862
Part of
KoMaL A Problems 2023/2024
Problems
(1)
Quadrilateral geo with incircles
Source: KoMaL A862
11/11/2023
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in circle
ω
\omega
ω
. Let
F
A
,
F
B
,
F
C
F_A, F_B, F_C
F
A
,
F
B
,
F
C
and
F
D
F_D
F
D
be the midpoints of arcs
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
of
ω
\omega
ω
. Let
I
A
,
I
B
,
I
C
I_A, I_B, I_C
I
A
,
I
B
,
I
C
and
I
D
I_D
I
D
be the incenters of triangles
D
A
B
,
A
B
C
,
B
C
D
DAB, ABC, BCD
D
A
B
,
A
BC
,
BC
D
and
C
D
A
CDA
C
D
A
, respectively.Let
ω
A
\omega_A
ω
A
denote the circle that is tangent to
ω
\omega
ω
at
F
A
F_A
F
A
and also tangent to line segment
C
D
CD
C
D
. Similarly, let
ω
C
\omega_C
ω
C
denote the circle that is tangent to
ω
\omega
ω
at
F
C
F_C
F
C
and tangent to line segment
A
B
AB
A
B
. Finally, let
T
B
T_B
T
B
denote the second intersection of
ω
\omega
ω
and circle
F
B
I
B
I
C
F_BI_BI_C
F
B
I
B
I
C
different from
F
B
F_B
F
B
, and let
T
D
T_D
T
D
denote the second intersection of
ω
\omega
ω
and circle
F
D
I
D
I
A
F_DI_DI_A
F
D
I
D
I
A
. Prove that the radical axis of circles
ω
A
\omega_A
ω
A
and
ω
C
\omega_C
ω
C
passes through points
T
B
T_B
T
B
and
T
D
T_D
T
D
.
geometry
komal